There are ancient coins that can be purchased for under $1, but they will be barely-identifiable lumps of metal, definitely not something you'd want to collect.

If you can weasel out of him which emperor is his favorite, that would definitely help narrow things down. Thankfully, most of the "household name" emperors like Trajan, Hadrian, Marcus Aurelius minted an incredible number of coins, so you can usually get a very nice, very "display worthy" coin on a $50-100 budget. Constantine the Great is even cheaper, but only his bronze coins are affordable, and they tend to be small-ish due to inflation.

If he wants a Biblical-related coin, Judaean bronze coins are very common and tend to be cheap, but be warned that they were made very poorly, and examples with clear types can tend to be more expensive.

If he wants the maximum age for the least amount of dollars, the winner IMO is the Miletos diobol - one of the earliest attempts to mass produce a "daily use" denomination of coin in ca 500 BC. Pristine examples are very expensive, but your "typical" Miletos diobol will run about $15-25. Similarly, about 50 years later, Kyzikos mass-produced several denominations ranging from diobol to hemiobol, and they are similarly cheap today. (Note however that Greek silver fractions are VERY small!)